The Reconstruction Conjecture

Let G be a graph on at least three vertices
and v be a vertex of G.
Then G-v is a vertex deleted subgraph
of G.
The collection of vertex deleted (unlabelled) subgraphs of G,
together with their multiplicities,
is the deck of G.
Thus, the deck for a graph on n vertices consists of
n graphs, each of which has n-1 vertices.
Each of graphs in this deck could be called a card.

Given a deck of a graph G,
certain parameters of G are easily determined.
G has one more vertex than any graph in the deck.
Every edge of G is missing from exactly two cards,
thus each edge shows up on n-2 cards.
The number of G can be determined by
summing the numbers of edges on each card and
dividing the total by n-2.

Similarly, for any graph H
on fewer vertices than G,
the number of subgraphs of G
which are isomorphic to H
can be determined by summing the numbers of
such subgraphs on each card and
dividing the result by
|V(G)|-|V(H)|.
(Kelly's Lemma)

A parameter of G which can be deduced
from the deck is called
reconstructible.
Since, the number of edges of G is
reconstructible, the degree sequence of G
is reconstructible. Thus, if G is a regular graph,
G itself is reconstructible.

Disconnected graphs are reconstructible.
Trees are reconstructible.
Separable graphs without endvertices are reconstructible.

In 1967, Tutte proved that the
dichromatic (rank)
and Tutte (dichromate) polynomials
are reconstructible. Years later, he heard in a lecture that the
characteristic
polynomial and chromatic number were not known to be reconstructible.
The chromatic polynomial, chromatic number,
the flow polynomial, and the number of spanning trees
are easily deduced from the dichromatic polynomial; and Pouzet had shown that the only missing step in reconstructing the characteristic polynomial was counting the number of hamilton cycles. Tutte also showed how to count the Hamilton cycles.
A similar polynomial, called the
idiosyncratic
polynomial is also reconstructible.
In order to present the result in a way more people would understand, Tutte wrote a new paper in 1979. A few years later, W. Kocay gave a very simple proof using
Kelly's Lemma, and I will try to outline that proof
below, much as Tutte does in his 1998 book.

A.J. Schwenk (Almost all trees are cospectral.
New Directions in the Theory of Graphs.
Academic Press, 1973, pages 275-307.)
proved that, for almost every tree,
there is another tree
with the same characteristic polynomial.
Such pairs of trees must also have the same Tutte polynomial.
Thus, the Tutte polynomial and the characteristic polynomial
together are not enough to characterize a graph.
Here's one pair of co-spectral trees (Marshall, 1971) given as edge lists:
12,13,14,15,16,67,78 and 12,13,14,15,56,57,58.
(Trees are reconstructible.
It would be better to have examples of cospectral 2-connected graphs with the same rank polynomial.)